Modulus of a complex number is its absolute value i.e. if #z=a+ib#, #|z|=sqrt(a^2+b^2)# and
#argz=theta# if #z=|z|(costheta+isintheta)# and hence #argz=tan^(-1)(b/a)#
also note that using De Moivre's theorem #arg(z_1z_2)=argz_1+argz_2#
as if #z_1=r_1(costheta_1+isintheta_1)# and #z_2=r_2(costheta_2+isintheta_2)#
#z_1z_2=r_1r_2((costheta_1costheta_2-sintheta_1sintheta_2)+i(sintheta_1costheta_2-costheta_1sintheta_2))#
= #r_1r_2(cos(theta_1+theta_2)+isin(theta_1+theta_2))#
As #z_1=(sqrt3-1)+i(sqrt3+1)i#
now #|z_1|=sqrt((sqrt3-1)^2+(sqrt3+1)^2)#
and #argz_1=tan^(-1)
= #sqrt8=2sqrt2#, we get
#z_1=tan^(-1)((sqrt3+1)/(sqrt3-1))=(5pi)/12#
see details here
Similarly as #z_2=-sqrt3+i#, #argz_2=tan^(-1)(-1/sqrt3)=(5pi)/6#
Note #|z_2|=2#
Hence #arg(z_1z_2)=(5pi)/12+(5pi)/6=(5pi)/4#
Check #-># #((sqrt3-1)+i(sqrt3+1))(-sqrt3+i)#
= #-3+sqrt3+(sqrt3-1)i+i(-3-sqrt3)+i^2(sqrt3+1)#
= #-3+sqrt3+(sqrt3-1-3-sqrt3)i-sqrt3-1#
= #-4-4i#
= #4sqrt2(-1/sqrt2-1/sqrt2i)#
= #4sqrt2(cos((5pi)/4)+isin((5pi)/4))#
also #|z_1z_2|=4sqrt2=|z_1||z_2|#
